Exact enumeration, minimal interfaces, and universality for the(2,1)-gapped consecutive-ones property
Year: 2026
Type: preprint
Abstract: We establish an exact constructive theory for the (2,1)-gapped consecutive-ones property, us ing the number of internal zeros as the conserved quantity. The principal new result is a polynomial-time algorithm that counts all legal labeled-column orders of a laminar row fam ily, computes their complete total-defect distribution, and supports exact uniform sampling through bijective unranking. No bound is imposed on the number of rows, row sizes, hier archy depth, column frequency, or incidence width. Three boundary polynomials per cluster suffice: its internal orders, its responses to one internal foreign marker, and its responses to a marker immediately before its last column. A crossing lemma forces interactions between disjoint children to form a matching; the crossing weight factors as twice the product of two endpoint responses. This makes the composition polynomial. A rank-three context matrix proves that three coordinates are necessary within the exact linear boundary-representation model. The same interface transfers to a validated hierarchy of arbitrary local cores, giving an FPT algorithm in the maximum core size without forcing the cores to remain consecutive. A complementary theorem treats repeated column types without assuming laminarity. For fixed row types and budgets, the complete defect enumerator has an exact finite run factorization. Once every type multiplicity exceeds the sum of the budgets of rows omitting it, the number of labeled orders divided by the factorials of type multiplicities is an ordinary multivariate polynomial. Its degree is at most min(r+K,2r+K+1-d), where r is the number of distinct nontrivial rows, K their total budget, and d the number of present column types; the bound is sharp in general. The same strict threshold makes budget-feasibility equivalent to ordinary C1P. Both conclusions can fail at equality. A marked finite automaton supplies a second exact compiler, reusable inventory queries, and independently checkable counting and infeasibility certificates. The earlier universality results remain part of the contribution. Actual unit-budget matrix rows with nested interfaces represent every upward-closed NP predicate with a polynomial time verifier. For every fixed r≥2, recognizing whether all infeasible subfamilies of nested added rows have obstructions of size at most r is Π₂ᴾ-complete, even with a feasible base and prescribed low-order compatibility data. The original NP-completeness reduction is indepen dently audited and compressed; labeled-order counting is #P-complete under polynomial-time Turing reductions with unequal gadget multiplicities handled explicitly. The release includes complete central proofs, exact implementations, independent direct zero and symbolic-certificate checkers, exhaustive small-instance comparisons, executed large structured examples, and primary-literature boundaries. The new positive theory exploits observable boundary responses instead of enumerating internal orders. It supplies a constructive counterpart to the representation and recognition barriers, without asserting that arbitrary difficult instances admit small profiles.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225010721